#Euclidean construction

[charliecraftsthings]

My first ever attempt at sashiko!  Natural linen thread on denim jeans (left leg), shippo tsunagi pattern.

I thought this pattern would be a good beginner choice, but I was ambitious. I didn't care strongly about making even stitches, I just wanted to try this out.  

I created the grid circle-by-circle (compass and straightedge), using a very cheap drawing compass with a pen taped to it. This turned out to be a TERRIBLE mistake: 

Fabric stretches as you drag a pen over it (Euclidean construction on not-very-Euclidean surfaces, a bad idea--who knew)

As you move farther from the origin, the grid becomes more and more distorted as the errors compound

Since the pants are intact, placing rulers over seams (bumpy!) will distort measurements.

I also didn't take into consideration that at the intersections, I'd have to choose how threads would overlap. So there are some messy intersections.

But I sure learned a lot from this experience! 😅

[spoopy-moose]

So sad that Kentucky route zero never took off on tumblr because it’s so hard to fandomfy such a work because I have so many words of analysis on this game I love it so much

[notreallyherehahaha]

Euclidean Construction of the Golden Rectangle, Golden Triangle, Golden Gnomon, and Regular Pentagon

In this construction, the points used are in alphabetical order. The golden rectangle is rectangle BKOF, and the golden triangle is triangle FPB. There are two golden gnomons: triangle QFP, and triangle PBR. The regular pentagon is BRPQF. Every circle used, even just to bisect segments, is shown.

[wrong-directions]

Non-non-Euclidean Biblically Accurate Platonic Solid

[ckret2]

At some point, the Axolotl must've witnessed the aftermath of the Euclidean Massacre.

As you can see, Bill is very happy and definitely not at all traumatized and doing great and look at all these followers he's found who are definitely alive.

Here, have a fic about the Axolotl, the birth of the Nightmare Realm, and Bill trying so so hard to convince himself that he's the hero.

####

To the mortals he swam past, with their different calendars and their different ways of perceiving time, the great Axolotl's migration through space and between dimensions was an event of great note: his passing marked eclipses, tsunamis, festivals, omens, meteor showers; his migration was studied by astronomers and his position was marked in astrological birth charts.

To the Axolotl, he was on his daily commute home. He could take an interdimensional portal, but swimming was better for the environment and he could use the exercise.

He passed by the same two dimensional wall every day. It was covered with many little worlds, and so many of them populated with little mortals, and he'd never paid any particular attention to the wall—until yesterday. A bold little triangle had shouted at him as he passed. It had been an amusing conversation—first contact was always fun—but he'd been busy and couldn't talk more than a moment, just long enough for the Axolotl to be charmed that a lower-dimensional creature had yelled at him and for the triangle to be shocked that a higher-dimensional creature had answered. The triangle had told him that, to his two-dimensional people, these shadows on the wall, the Axolotl was an eclipse: they marked the time by the shadow he cast on their flat world during his commute.

He hadn't even learned the triangle's real name. The triangle had refused to tell him, instead introducing himself as the "Magister Mentium." Teacher of minds? Maybe it was a job title.

Between the nightmare of a case the Axolotl was currently handling and the fact that he'd had to stay late working, he'd nearly forgotten about yesterday's fascinating little meeting until he was leaving on his nightly commute. He didn't know how long the tiny shapes' life cycles were; he hoped the little triangle was still alive today. If not, maybe he'd left behind descendants.

But when he came up to the wall, it was gone.

The vacuum reeked of burning hydrogen.

The Axolotl stopped, puzzled. The wall wasn't empty, wasn't damaged, wasn't going through heat death—the entire thing was missing. No rubble. Surely it hadn't been demolished for some new construction? It had been in good condition. It was a fairly new plane of reality, likely under fifty billion years old. And it had admittedly been a few eons since the Axolotl had studied dimension use & zoning law, but last he checked it was unlawful to demolish a populated dimension without transplanting the growths first—which took much longer than a day. So what could possibly have done this? And what he saw behind the wall...

Something was very wrong. He started moving again, faster, looking for someone who could tell him what was happening. He kept the ragged rip in reality left by the missing wall in his peripheral vision. Stars and stardust slowly fell in, sucked through the tear. The wall must have come down by accident.

Nobody would have knowingly left behind such a large hole to Dimension Zero.

Assuming he was looking at Dimension Zero; he wasn't sure he was. Beneath all other dimensions was supposed to be a void, an empty in-between space. The zeroth "dimension" was simply reality's center point, the not-dimension between all dimensions; it wasn't a place. But with the two dimensional wall gone, he didn't see reality bending in toward a point like he should. He saw a roiling, nauseating mass of blinding colors, thrashing around each other like a frightened pile of injured worms.

Far in the distance, a full reality away, he saw a faint line of blue light.

It was several minutes before he began to run into other people. He passed a crew of cosmic firefighters and their ships, spread out over a span of space wider than an asteroid belt. The fact that they didn't appear to currently be fighting any fires was more disconcerting than a full blaze would have been. An eerie tension hung thick over the scene like invisible smoke. As the Axolotl swam by a couple of firefighters, he overheard them saying, "... orders of magnitude higher than anything we've been trained to handle. An entire reality catching fire is one thing, but the concept of realitycatching fire...?"

"And the speed it's moving..."

"Excuse me," the Axolotl said, trying to keep the edge of fear out of his voice. (Why was he so afraid? He was barely acquaintances with one resident on the wall.) "Can you tell me what happened to the wall? It was just here yesterday."

Rather than explain, one of them pointed in the direction he'd been going. "Sorry, we don't know any more than you do. Look for the storm. You can't miss it."

The other asked, "Are you one of the guys with the apoc cops?"

His fear leaped higher. The "apoc cops" were members of the Apocalyptic Threat Task Force. "No. Sorry, I have to go." He swam onward toward the blue line of light.

The stench of burning hydrogen grew stronger. He smelled something else acrid underneath.

####

To his slight relief, the "storm" wasn't the disaster that had brought down this wall. Rather, it was a person: a lightly raining storm cloud with a gray rain-soaked fedora perched on top, hovering in space.

It was talking to a hapless-looking furred serpent twice the Axolotl's length with four mismatched limbs: she clutched a can of spray paint in her claws, and was so nervous he could hear the marble in the can rattling. A disembodied sunbeam pierced the eye of the storm cloud to shine in the serpent's face as she spoke, and a tornado swirled beneath its cloud, carrying all its personal effects—including a tumbling badge from the Apocalyptic Threat Task Force, its logo of a mushroom cloud struck out with the "no" symbol still visible through a thin glaze of sleet. A chill ran through the Axolotl at the sight of that badge.

The cloud wasn't the only one with the apoc cops on the scene. There were several other investigators nearby, taking readings where the wall used to be. The Axolotl didn't like just how many were buzzing around. They seemed far too busy for far too empty a space, and they steered far too clear of the thrashing, multicolored miasma covering the emptiness that should contain Dimension Zero.

There were several stars in the area that the investigators had to work around. Between the crowds and the missing wall, it took the Axolotl a moment to realize where they were: this was the spot he'd met the triangle yesterday. He was sure of it. He recognized the star right next to the missing wall, the one the triangle had told him he eclipsed during his commute. He'd passed it millions of times.

Why had the apoc cops set up here?

The star was slowly falling toward the roiling miasma where Dimension Zero should have been. He nudged it back into place as he passed.

As the Axolotl approached the duo, the serpent was saying, "I told you, I don't know how it caught fire! I was just passing by..." The storm cloud's sunbeam dropped from her face to point skeptically at her spray paint. She hid it behind her back and quickly went on, "I was just passing by, minding my own business and not doing anything illegal, and suddenly the whole wall went up in flames!"

The cloud said, "The whole wall? Simultaneously?"

"The whole thing! I mean... it kind of rolled across the dimension, but—it took less than ten seconds to cover everything I saw!"

"Which direction did the fire travel?"

While the serpent tried to remember, the Axolotl swam up to the storm cloud. "Excuse me, the firefighters said you're in charge of the investigation?"

"Currently," the cloud said, in a tone that suggested it very much wished it wasn't. It looked over the Axolotl, then turned back to the serpent—she flinched when its sunbeam hit her face again—and it asked gruffly, "Is this your lawyer?"

The serpent looked hopeful. "Are you my lawyer?"

"No, I'm not," the Axolotl said, perturbed. Potential defendants aside, nobody ever insinuated he was somebody's lawyer and meant it in a nice way—and he was on the receiving end of such accusations more and more often lately. His reputation was beginning to precede him. "We've never met. I'm trying to find out what happened to this wall. I know a—friend in there. You said something about a fire?"

An active ATTF investigation was in no way the Axolotl's business. But people had a tendency to cooperate with professionals, whether or not their profession had anything to do with the situation at hand. The ATTF agent turned to the Axolotl and said, "You had a friend in there. The wall that used to be here, Dimension 2 Delta, has been completely incinerated."

The Axolotl stared at the cloud, trying to process that. But the whole wall had been there yesterday. Billions of galaxies, each with trillions of stars, each capable of supporting trillions of species—never mind lives. "You can't mean completely. Surely there are some survivors?"

"Not a single one," the cloud said. "Not even gods and ghosts made it out."

"How?"

"That's what we're trying to figure out," the storm said. "Right now, the only witness we've found was the person who called in the emergency." A branch of lightning pointed toward the serpent. "And she doesn't know a damn thing." The serpent nodded in enthusiastic agreement.

"But that's... How does an entire dimension disappear with only one witness?"

"Very quickly," the storm said. "The apocalypse Origin & Cause investigation can't make heads or tails of the scene—" a gust of wind swept demonstratively toward the other apoc cops taking readings near the missing wall, "but far as we can tell, the damn thing spontaneously combusted—somewhere near here."

The Axolotl stared helplessly between the serpent and the storm. "Dimensions aren't supposed to spontaneously combust," he said, very reasonably and very unnecessarily.

"Tell 2Δ that," the storm said. "Only time a dimension moves that fast is during a Big Bang explosion or a Big Crunch implosion—and 2Δ wasn't undergoing a Big Crunch. No natural one, anyway. In all my eons with ATTF, I've never seen anything like it."

The Axolotl had been around enough eons himself to know that, after a certain point, novelty became very, very scary—because things working like they should shouldn't do anything you'd never seen before. He worriedly searched the roiling chaos exposed by Dimension 2 Delta's collapse for any sign of what had happened.

The chaos simply thrashed. It moved like it was in pain.

"Did that..." the Axolotl gestured vaguely toward the chaotic foam, "have anything to do with the wall's combustion?"

The serpent shrugged. "I didn't see it until after the fire went by."

The storm grunted uncertainly, a low, thunderous grumble. "Heck if we know. It's connected, no doubt about that—but we haven't even figured out what it is yet. All we know is, it shouldn't have been behind the wall."

The Axolotl stared into the roiling colors, looking for anything visible through the thrashing kaleidoscopic colors.  "If you don't know what it is yet—then, how do you know there aren't survivors in there?" The Axolotl couldn't stop seeing that poor, frightened, awed triangle he'd met yesterday. All the people who'd once been in Dimension 2 Delta mattered—of course they did, those billions of trillions of trillions of billions of lives; he wanted any of them to survive—but that triangle was the one he knew, the one he saw in his mind's eye now. The whole dimension was contained inside that triangle. He had to hope. "I'm going to check."

"What—? You're crazy! Don't you know falling into Dimension Zero will destroy you?!"

"I know falling into Dimension Zero destroys you; I don't know what falling into that thing will do." He squared up with the chaos and steeled his nerves. "Besides, I can regenerate. I'm an axolotl."

"But—!"

"Sorry, there isn't time for more questions." He swam into the maelstrom.

####

Dimension Zero was supposed to be a singularity. Like a black hole, but even smaller—a point so dense it broke physics. If you fell in you'd be crushed into that point by the weight of all realities, a point so small it had no volume.

But whatever was behind where the wall had been, it was certainly no point.

As soon as he crossed the threshold, he was barraged with a psychic hurricane. Reality frothed and foamed like a flood spilling from a burst dam. Distant baby stars were born and popped like bubbles, and old stars fell in and were gloriously reignited. His every sense was bombarded with infinite sensations—every color and image in this dimension all at once; every song that had ever been played playing in the same instant and the instant extended indefinitely; strobe lights that were both flashing on and flashing off at the exact same moment. Beneath the music was a constant hiss like the background radiation of reality, the static echo of a universe's birth, but much too loud; he could swear it sounded like gibbering, babbling voices, their desperate messages unintelligible. He smelled every scent, including the lingering smell of burning hydrogen that he'd noticed outside; but above and beyond all that, he smelled the stench of burning life.

He knew now, this was Dimension Zero: it was as if all of spacetime had been crushed into a singularity, but then the singularity was bloated up to the size of an entire universe. Dimension Zero was never supposed to be this bloated.

And the most terrifying part: there were people in this bizarre ruin of a dimension. Millions of them. (Just as horrifying: there were only millions of them.) He was sure he must have been hallucinating—here, dreams and reality swirled around each other like a bottle of water and oil shaken until they were forced to mix—but the longer he looked, the more sure he was that the people were a part of reality. They were, perhaps, the most real thing in the entire dimension.

They were all dancing.

They were all dead.

"Heeey, look who's here!" Suddenly, in front of the Axolotl, there he was—as if he'd always been in front of the Axolotl, as if he were always everywhere at once. The ghost of the little triangle he'd seen yesterday, neon incorporeal. "Happy New Year, everybody!" He laughed. "Get it? That—that's a joke, time doesn't pass in the dream realm, so..." The triangle waved off the Axolotl. "Oh, you wouldn't get it. Screw you. Anyway, introductions! I should do that." 

The triangle was extremely inebriated. He was blinking blearily, floating crookedly, moving in odd uncoordinated jerks, his pupil expanding and contracting with no correlation to the light it was taking in. He seemed to flicker across multiple timelines that had been collapsed into one, like a drunk that couldn't walk a straight line: appearing here then there, then multiple places at once, then everywhere; and then became everywhere, and then collapsed again to a single triangular point. The Axolotl had the worrying impression that the triangle hadn't been sober for a long time.

"So! These are my people!" He gestured with a flourish to the dancing corpse puppets. The strobe lights—which, the Axolotl only now realized, didn't actually have a source, but were rather disembodied rays of light emanating from nothing—turned to highlight them from every angle. It was like a cloud of glitter, all these tiny, flat, jewel-tone flecks, emerald and citrine and ruby and sapphire, triangles and squares and pentagons and hexagons. Each with two spindly arms; some with legs and some without; a single dull eye or a slack mouth; some of them cracked and chipped like broken glass, some of them crushed and melted together into multi-corpsed horrors, some of them fraying and peeling apart around the edges like fabric; so much silvery blood dripping and floating around them. Such beautiful, colorful dancing gore. "All my followers and friends! They love me! They couldn't see you last time you flew by, but thanks to me, they sure can now! Say hellooo!"

It took the Axolotl a moment to realize that the triangle's eye was boring into him and the instruction was for him. "Hello," he said weakly. 

"Very nice." The triangle turned without turning to the millions lost inside Dimension Zero, reality shifting around him to put all of the dimension's prisoners in front of his eye. The Axolotl reeled from existential vertigo. "Now check this out!" The triangle gestured at the Axolotl for his people's benefit. "Behold! Your Magister Mentium presents to you: the eclipse! In the horrifying pink flesh! Quite a sight, huh?"

Many of the dancers turned toward him. Some aimed their dull, dead eyes in his direction. He shivered under their chill stares.

Heedless of the Axolotl's horror, the triangle elbowed him. "I didn't peg you for a party crasher, pinky!" (The triangle's touch was so cold.) "But hey, the more the merrier. Welcome to the dream realm, have a drink!"

A 2D cup manifested in front of the Axolotl that, based on its smooth, featureless yellow surface and its glow, appeared to be made from the triangle's own ghostly flesh. It seemed to be filled with watered-down raw existence. He didn't touch the cup. "What's the dream realm?" He couldn't stop staring at the dancers macabre.

"This is!" The triangle stretched out his arms—and stretched them, and stretched them, seeming to embrace all of reality at once. The Axolotl got the terrifying impression he was within the embrace too. "The realm of dreams! My realm! Paradise of color and light! Realm of spirits and muses!"

"It looks more like a nightmare."

"Do I come to your house and insult your wallpaper? Buzz off."

When the triangle dismissively floated away from him, the Axolotl again got the dizzying sensation that he was the one moving. The truth finally dawned on him:

The triangle, somehow, was literally the center of this universe. Point 0,0,0 on the cartesian plane of reality. Whenever he moved, Dimension Zero moved with him. When he backed away from the Axolotl, Dimension Zero backed with him, rushing past while the Axolotl held still.

And not once during their conversation did any of the millions of dead shapes stop dancing. 

"What are you doing?" the Axolotl asked, voice hushed.

"Partying," the triangle said. "We're having a party."

The Axolotl couldn't tear his eyes from the choreomaniacs' forced revelry. "How long have you been partying?"

"Uhh... pfff... I dunno, hard to keep track. A few months?" The triangle turned toward his tortured people. "Hey! How long have we been partying?"

One of the bodies mixed in amongst the dead, boogying deliriously, faintly cried back, "Time has no meaning and eternity has collapsed into a single unending moment of bliss!" (The Axolotl shuddered at the grotesque ventriloquism act.)

"Oh, yeah, right, forgot I decreed that. Thanks, pal!"

"You're welcome, oh wise and glorious Magister Mentium!"

The triangle turned back to the Axolotl. "An eternity."

The Axolotl tore his horrified eyes away from the dancers. "What about all the others?"

The triangle paused. "I don't know who you're talking about." The background radiation hissed in agitation.

The Axolotl very much suspected he did. "Your other people."

"There aren't any others," the triangle said defensively.

"There were! All of the other shapes around your world! All of the lives on other worlds! Where are all those people?!" He hoped that they might have gotten evacuated to a neighboring wall, or that they'd been concealed somehow, or even that they'd been collapsed together into the shapes he saw before him and could still be separated—

"It's fine," the triangle said stiffly. "Nothing important was lost."

"Nothing important?" the Axolotl repeated, shocked. "This was an entire dimension—!"

"A wall," the triangle said.

"A wall with lives on it—"

"Shadows."

"And do shadows not deserve to live?!"

The triangle flinched at the question as his good cheer crumbled. He didn't answer, but he gave the Axolotl a heavy, hard, emotionless look—a wretched, empty look—and the Axolotl knew he knew they did deserve to live.

"They don't matter," the triangle lied. "Nothing important was lost. Only the true believers and the worthy remain."

"Your dimension had billions of trillions of stars alone. All the people surrounding them—"

"I didn't see any stars!" He said it so vehemently—as though, if he didn't see them, they must not have existed. As though he refused to acknowledge their existence. "I told everyone about the third dimension, I told them we were going, they had their chance to join me!" His voice was shaking. As he spoke he grew larger, until he was as large as the Axolotl—or perhaps the universe had contracted around him. "And if they refused to join the liberation, then they are what we liberated ourselves from!" Distant bolts of lights flashed through Dimension Zero, responding to the triangle's outrage; the nearest stars blazed brighter for him. His dead people screamed in terror. They didn't stop dancing.

"You... tried to leave your dimension before the fire reached them?" Had he tried too late?

The triangle flinched again; his appearance flickered, like a TV that for a moment had picked up a pirate station broadcasting on the same frequency. The whispers hissing beneath the music grew more excited again, but the Axolotl still couldn't make out what they said beneath the party music.

The triangle said, "The... the fire came second."

"What came first?"

But he didn't answer. "Yeah, I brought them here." He spread his arms again, gesturing at the other shapes. "They followed me, and I freed them from our flat, restrictive dimension. They're all fine. And they all love me for saving them."

"Saving them?" he echoed. He wanted to laugh in disbelief, but it felt too much like laughing at a stranger's funeral. Laughing at an open mass grave. "But—everyone here is already dead. Even you." The triangle should be in an afterlife. Whatever afterlives his dimension once had, they were gone now. The Axolotl would have to help the triangle find one in another dimension—the paperwork alone would take time he didn't have to spare; he'd probably have to split off a timeline or two to squeeze it in...

The triangle snapped, "Whoa, hey, hey! Watch who you call dead, buddy! Look at me!" He stretched out his limbs, glowing dazzlingly bright. Brighter than a star. Even the Axolotl had to turn away from the blinding light. "I transcended my body! I'm made of pure energy! This is the most alive I've ever been!" A being of pure energy that had lost its physical form was the very definition of a ghost; but the Axolotl didn't have a chance to argue before the triangle went on, "And does anyone here look dead? Everyone's dancing! We're all having a great time, aren't we?" A few corpses groaned and gurgled in response.

If the triangle wanted to be a wandering ghost, fine. That was his prerogative. But he had no right to force the remains of his followers to deny their death with him. "Look—look at your people," the Axolotl commanded. "You're making them dance! You must know what state they're in!"

Without actually moving, the triangle had somehow become the space in between the Axolotl and his choreomaniacs, forming a sharp shield in between them. "You don't know what you're talking about. They're fine. They're immortal!"

The Axolotl gestured furiously past the triangle. "LOOK AT THEM!"

The triangle's gaze flickered toward them for a split second. The Axolotl saw guilt flashing in his eye; but then he squeezed his eye shut. "No, you look at them. Maybe it took me a little bit to get it right, but they're all great now."

To get it right? The Axolotl peered around the triangle at the shapes again, and only now saw that he was right.

Not all of them were dead.

Some were trapped in ecstatic trances; some were numb with terror; some were already long dead, and yet the corpses weren't being puppeted like he'd assumed—they danced under their own power. There were amalgams of a dozen, a hundred bodies fused together into shambling, gyrating horrors—but there was still life in their horrified eyes and their limbs twitched independently. The ones that were bleeding just kept bleeding and bleeding and bleeding, unending, blood never clotting nor running dry. The corpses and the comatose and the ailing and the bleeding dancing with the living that craved death.

The triangle was responsible for their condition?

He glided between the corpses, sliding his arms around a few of them. They kept dancing.  "I didn't quite get to a few of them in time, so I took the empty space where their souls used to be and filled them with an insatiable hunger to party," he said. "And look, they're good as new! Probably better than they were before, even!"

"These bodies should be laid to rest," the Axolotl said heatedly, "and the rest of you should be dead."

The triangle went still.

The Axolotl remembered, a second too late, that that was a perfectly normal thing to say to deceased clients and other gods in his line of work, but the kind of thing that scared the living daylights out of mortals.

"So that's a threat." His arms slid off the shapes; his fingers were stained with silvery blood that shimmered like static noise.

"No! No. But the condition that you're all in..."

"You'd better check yourself, frills," the triangle snapped. "You crash our party, in our eternal paradise, and start threatening us! Who the hell do you think you are, telling us we should be dead?!"

The Axolotl paused uneasily. "A fully licensed psychopomp...?"

"Well you'd better keep your psycho, pompous paws off my people!" The triangle blazed bright red, literally incandescent with rage. Some of his "people" slowly stopped dancing and turned their hollow eyes toward the Axolotl.

And the Axolotl couldn't say why, but he was suddenly sure he was in very grave danger.

He backed up from the triangle, moving in the direction that the edge of Dimension Zero should have been, although he was no longer sure whether it was still behind him. "I... think I should leave."

"I think you'd better."

He turned and fled. He couldn't explain his panic, but he felt in his bones like something was chasing him. He had to spend longer than he wanted searching for the edge of this bizarre reality—the triangle had turned and twisted and moved the borders so many times that he'd completely lost his bearings—spied the nearest exit, and darted for it between two unfinished planes of reality.

He thought he felt flames at his back.

The triangle's voice followed him out: "Next time, poop on somebody else's party!"

He tumbled through the membrane between the overbloated Dimension Zero and the higher dimensions with the relief of a suffocating fish escaping its net to plummet back into the water. He had to take a moment to reorient himself to his surroundings—time passing so that each moment took its turn and ended when it was over, space that felt like space rather than all distances collapsed in on themselves—and looked back at Dimension Zero.

The longer he stared into the kaleidoscopic miasma, the more sure he was that, no matter where he looked, right at the center of his field of view, he could always see a shining yellow fleck of triangular glitter.

"I'm sorry," he said. "I spoke out of emotion. I am glad that you—" well, "survived" wasn't the right word, "—still exist. And it was heroic of you to save as many people as you did. I shouldn't have said they shouldn't be alive; just..."

He felt like he could still see the shapes dancing in the corners of his eyes.

"... Just not alive like that."

####

Who was the triangle?

At their first meeting yesterday, it had been clear to the Axolotl that the triangle could see and perceive things off his wall while the rest of his people could not; he'd identified himself as "Magister Mentium" rather than by name; and he'd been surrounded by shapes, all turned toward him, listening: so perhaps he was a leader of some kind? He must have seen whatever destroyed their dimension coming and been able to use his position to evacuate a few people. The true believers and the worthy, he'd said—maybe his... congregation? Maybe he was a religious leader? At any rate, it was a miracle he'd saved as many people as he had with what must have been very short notice.

But... their forced dance... the bodies fused together... the living-who-should-be-dead bleeding and bleeding and bleeding without end...

The Axolotl didn't want to believe the triangle had any ill will. He reminded himself that he didn't know anything about his people or their culture. These shapes had been through something unimaginably traumatic. They'd watched an entire reality die; many of them were stuck in the process of dying in a place where they couldn't complete it. Any mortal would be insane with grief. Perhaps their magister was just leading them in some sort of cathartic dancing mania; perhaps this was how the shapes processed their grief. He hoped that was what it was. He hadn't gotten a chance to speak to the others—he didn't know how many could speak—but he had seen, for just a moment, how survivor's guilt ate at the triangle.

The storm cloud with the Apocalyptic Threat Task Force had said that every single living being from Dimension 2 Delta had been killed. Even the gods and the ghosts. So how had the triangle and his people survived?

And what were they doing here, in the singular heart of all reality?

And what had happened to their world?

####

(Hello, thanks for reading!! If you were lured in by the colorful art I laid out as bait and this is your first time here, welcome!! This is part 1 of a 5-or-6 part fic about the Axolotl in the immediate aftermath of the Euclidean Massacre. I'll be posting one chapter a week, Fridays 5pm CST, so stick around if you wanna read more and learn the exciting answers to exciting questions like "Bill where in the good goddamn did you find a bunch of half-dead shapes??"

It's ALSO chapter 61 of an ongoing post-canon post-TBOB very-reluctantly-human Bill fic. So if you wanna read more of me writing Bill, check it out here. If you're not sold on the idea of a human Bill fic, I've also got a one-shot about normal triangle Bill escaping the Theraprism if you wanna read that.

If this is NOT your first time here and you already knew all of the above: hey y'all remember when we had to skip over chapter 61 because it would've been posted like four days after TBOB came out and it needed MAJOR revisions? Well, here it is!! And also it's currently like six times longer than it was originally. We're gonna be hanging out with the Ax for like a month and a half, buckle up. 

Let me know what y'all think so far!!)

[haggishlyhagging]

It took about two hours for Daina Taimina to find the solution that had eluded mathematicians for over a century. It was 1997, and the Latvian mathematician was participating in a geometry workshop at Cornell University. David Henderson, the professor leading the workshop, was modelling a hyperbolic plane constructed out of thin, circular strips of paper taped together. 'It was disgusting,' laughed Taimina in an interview.

A hyperbolic plane is 'the geometric opposite' of a sphere, explains Henderson in an interview with arts and culture magazine Cabinet. 'On a sphere, the surface curves in on itself and is closed. A hyperbolic plane is a surface in which the space curves away from itself at every point.' It exists in nature in ruffled lettuce leaves, in coral leaf, in sea slugs, in cancer cells. Hyperbolic geometry is used by statisticians when they work with multidimensional data, by Pixar animators when they want to simulate realistic cloth, by auto-industry engineers to design aerodynamic cars, by acoustic engineers to design concert halls. It's the foundation of the theory of relativity, and thus the closest thing we have to an understanding of the shape of the universe. In short, hyperbolic space is a pretty big deal.

But for thousands of years, hyperbolic space didn't exist. At least it didn't according to mathematicians, who believed that there were only two types of space: Euclidean, or flat space, like a table, and spherical space, like a ball. In the nineteenth century, hyperbolic space was discovered - but only in principle. And although mathematicians tried for over a century to find a way to successfully represent this space physically, no one managed it - until Taimina attended that workshop at Cornell. Because as well as being a professor of mathematics, Taimina also liked to crochet.

Taimina learnt to crochet as a schoolgirl. Growing up in Latvia, part of the former Soviet Union, 'you fix your own car, you fix your own faucet - anything', she explains. 'When I was growing up, knitting or any other handiwork meant you could make a dress or a sweater different from everybody else's.' But while she had always seen patterns and algorithms in knitting and crochet, Taimina had never connected this traditional, domestic, feminine skill with her professional work in maths. Until that workshop in 1997. When she saw the battered paper approximation Henderson was using to explain hyperbolic space, she realised: I can make this out of crochet.

And so that's what she did. She spent her summer 'crocheting a classroom set of hyperbolic forms' by the swimming pool. 'People walked by, and they asked me, "What are you doing?" And I answered, "Oh, I'm crocheting the hyperbolic plane."' She has now created hundreds of models and explains that in the process of making them 'you get a very concrete sense of the space expanding exponentially. The first rows take no time but the later rows can take literally hours, they have so many stitches. You get a visceral sense of what "hyperbolic" really means.' Just looking at her models did the same for others: in an interview with the New York Times Taimina recalled a professor who had taught hyperbolic space for years seeing one and saying, 'Oh, so that's how they look.' Now her creations are the standard model for explaining hyperbolic space.

-Caroline Criado Perez, Invisible Women

Photo credit

[nateconnolly]

I'm going to be pretentious because it's fun for me personally. I'm not doing this as a gotcha; I'm doing it because I found the meme interesting, and I wanted to see how well it applies to Euclid.

My mathematical knowledge is almost exclusively Ancient Greek geometry, so I can't speak to non-Euclidean mathematics. But if we're going off the historical Ancient Greek definition from Euclid's Elements, the shape above is not a square.

The Elements is the world's oldest still-surviving long mathematical treatise. It begins with a series of definitions.

Def. 22 says

Of quadrilateral figures, a square is that which is both equilateral and right-angled...

So squares are quadrilaterals. Def. 19 says

Rectilinear figures are those which are contained by straight lines... quadrilateral those contained by four...

If quadrilateral is a type of rectilinear figure, then squares have to be rectilinear. And rectilinear figures are contained by straight lines.

Alas, the construction in the diagram is not made of straight lines, so it's not a square. (Henceforth, I'm going to refer to that shape as meme-pseudo-squares.)

BUT. And here's where shit gets interesting. You could use meme-pseudo-squares to do the Pythagorean theorem. The formula "a² + b² = c²" can be represented geometrically with literal squares on the lines of a triangle.

If you construct three meme-pseudo-squares with areas equal to square AZBH, square BGDE, and square KGAQ, then you could name the three meme-pseudo-squares A, B, and C

And the formula "a² + b² = c²" would still be true.

You would not be able to begin with meme-pseudo-squares and follow the steps in the Elements that prove Prop 1.47, unless you assumed as a postulate that you can construct meme-pseudo-squares with area equal to any given area, which feels like cheating. The only shapes you can create as postulates in the actual text are straight lines and circles. Incidentally, you can easily draw pretty good diagrams of those shapes with two sticks, some rope, and dirt. I don't think that holds for meme-pseudo-squares.

But the enunciation of Prop 1.47 is still true if you rephrase as

In right-angled triangles the meme-pseudo-square on the side opposite the right angle equals the sum of the meme-pseudo-squares on the sides containing the right angle.

Once you proved 1.47 and the proportion stuff in Book V, you could double back and demonstrate the indented claim above.

[fjordfocused]

and if nothing else, my liege, I hope my non-euclidean vocabulary fucked your sentence construction to smitherbeans

[mathematical-cheese]

🌻 :3

I will tell you& about a cool topology fact that uses one of my favourite theorems!

First, a primer of finitely presented groups:

Given a finite set with n elements S={a₁,...,aₙ}, we define a word to be a finite concatenation of elements in S. For example, a₁a₇aₙ is a word. We define the empty word e to be the word containing no elements of S. We also define the formal inverse of the element aᵢ in S, written aᵢ⁻¹, to be the word such that aᵢaᵢ⁻¹=e=aᵢ⁻¹aᵢ, for all 1≤i≤n.

We define the set ⟨S⟩ to be the collection of all words generated by elements of S and their formal inverses. If we consider concatenation to be a binary operation on ⟨S⟩, then we have made a group. This is the free group generated by S, and is called the free group generated by n elements.

Some notation: if a word contains multiple of the same element consecutively, then we use exponents as short hand. For example, the word babbcb⁻¹ is shortened to bab²cb⁻¹.

Note: concatenation is not commutative. So ab and ba are different words!

We now define a relation on the set ⟨S⟩ to be a particular equality that we want to be true. For example, if we wanted to make the elements a and b commute, we include the relation ab=ba. This is equivalent to aba⁻¹b⁻¹=e. In fact, any relation can be written as some word equal to the empty word. In this way, we can view a relation as a word in ⟨S⟩. So we collect any relations on ⟨S⟩ in the set R.

Finally, we define the group ⟨S|R⟩ to be the group of words generated by S subject to the relations in R. This is called a group presentation. An example is ⟨z,z²⟩, which is isomorphic to the integers modulo 2 with addition ℤ/2.

If both S and R are finite, we say that ⟨S|R⟩ is a finite group presentation. If a group G is isomorphic to a finite group presentation we say G is a finitely presented group. It is worth noting that group presentation is by no means unique so as long as there is one finite group presentation of G, we are good.

In general, determining whether two group presentations is really really hard. There is no general algorithm for doing so.

Lots of very familiar groups of finitely presented. Every finite group is finitely presented. The addative group of integers is finitely presented (this is actually just the free group generated by one element).

Now for the cool topology fact:

Given a finitely presented group G, there exists a topological space X such that the fundamental group of X is isomorphic to G, i.e. π₁(X)≅G. This result is proved using van Kampen's Theorem which tells you what happens to the fundamental group when you glue two spaces together.

The proof involves first constructing a space whose fundamental group is the free group of n elements, which is done inductively by gluing n loops together at a single shared basepoint. Each loop represents one of the generators. Then words are represented by (homotopy classes of) loops in the space. Then we use van Kampen's Theorem to add a relation to the fundamental group by gluing a disc to the space identifying the boundary of the disc to the loop in the space that represents the word for the relation we want. We do this until we have added all of the relations we want to get G.

We can do a somewhat similar process to show that any finitely presented group is the fundamental group of some 4-manifold (a space that locally looks like 4-dimensional Euclidean space, the same way a sphere locally looks like a plane). This means that determining whether two 4-manifolds are homeomorphic or not using their fundamental groups is really hard in general because distinguishing finitely generated groups is hard in general.

P.s. I also want to tell you that you're really wonderful :3 <2

[rlyehtaxidermist]

a short guide

euclidean:

a line segment may be constructed between any two points

a line segment may be extended infinitely

a circle may be made centred on any point with any radius

all right angles are equal

if a line crosses two other lines such that the angles they form on one side are less than right angles, the crossed lines intersect on that side

not euclidean:

the surfaces of any finite object

city blocks

pretty much every way you've actually interacted with notions of distance in your life

[lipshits-continuous]

Topological Spaces 2: Topologies and Continuity

Welcome to the second post in my introductory series to topology! The goal today is to give the definition of both topological spaces and continuous maps between them. These definitions will be very familiar from the last post so I highly recommend reading it here.

I do make brief reference to some constructions from linear algebra and abstract algebra however these aren't necessary for understanding the post. They are merely inserted for futher emphasis for those who are familiar.

2.1: Topological Spaces

In the last post, we saw that open sets are potentially a structure that we wish to study. Our definition will be in terms of properties we wish open sets to have without invoking any notion of a distance between point. Fortunately, open sets in metric spaces have properties which fit this exactly in the form of Lemma 1.11. We shall use these to give our definition of a topological space!

Definition 2.1:

Remarks:

We can show by induction that finite intersections of open sets are open. This property is equivalent to (T3), but (T3) is usually an easier property to check.

We can have different topologies on the same set as we shall see in the examples!

Examples 2.2:

Remarks:

The topology on ℝ induced by the absolute value metric is called the standard topology on ℝ. Similarly, The topology induced by the Euclidean norm on ℝⁿ is called the standard topology on ℝⁿ.

The discrete topology is actually the same topology as the topology induced by the discrete metric. The proof involves showing that every singleton set (set that contains one element) is open in X with the discrete metric. Then use Lemma 1.11 to conclude that every set is open in X with the discrete metric.

Just like how some metrics can't be realised as the metric induced by a norm (and how some norms can't be realised as the norm induced by an inner product), not all topologies can be realised as the topology induced by a metric. A topology which can realised as the topology induced by a metric is called metrizable. The area of study dedicated to classifying metrizable spaces is called metrization theory and is very rich (and very much beyond the scope of these posts)

Both the indiscrete topology and the topology in 4) are non-metrizable. We shall see a proof of this in a future post.

One thing we may want to do is to make new topological spaces out of old ones. We don't yet have enough theory developed to do much but one easy construction we may do is putting a topology on subsets of our topology space:

Proposition 2.3:

2.2 Continuous Maps:

Now we have defined what a topology is, we may finally generalise the notion of continuity! Our guide will be Lemma 1.12.

Definition 2.4:

Remarks:

Continuity very much depends on the topology.

This really is a generalisation of continuity since Lemma 1.12 says that for metric spaces this notion is equivalent to our ε-δ definition (Definition 1.3)

We shall now see some examples in the form of the following proposition:

Proposition 2.5:

Now we shall define a special kind of continuous map: homeomorphisms. You may have heard of these from pop maths, especially in the context of a coffee mug and a donut being "the same" to a topologist.

Definition 2.6:

Remarks:

Not only are homeomorphisms bijections between the sets, they also induce a bijection between the topologies:

This means that in some sense the topologies on X and Y are the same!

More formally, one can show that ≅ is an equivalence relation on topological spaces. The proof relies on 1) and 2) from Proposition 2.5

Homeomorphisms are in this sense "topological space isomorphisms" since they preserve the important structure of topological spaces: the topology. This is analagous to how bijective linear maps are vector space isomorphisms and how bijective group homomorphisms are group isomorphisms.

Example 2.7:

In fact, we may show any open interval (a,b) is homeomorphic to ℝ by showing that (0,1) is homeomorphic to (a,b) via an affine map then using the fact that "being homeomorphic" is an equivalence relation.

One major goal of topology is to classify topological spaces up to homeomorphism. This is insurmountable if attempted all at once, even if one restricts to special topological spaces, and is still an active area of research. What is a more approachable goal is to find certain properties that topological spaces have which are invariant under homeomorphism. A vast majority of what follows in this series of posts can be seen as a hunt for these invariants whilst also generalising a lot of familiar concepts we have on ℝ with its standard topology.

[charliecraftsthings]

My second attempt at sashiko.  Natural linen thread on denim jeans (the same pants), shippo tsunagi pattern.  

Hey so remember how I said creating a grid by circle-by-circle on fabric was a terrible idea?  I decided to do it again anyway 🤦

On this leg, I didn't make the edges of my design straight; I wanted it to look like circles were almost randomly selected to be in the pattern or not.  I also added colourwork, because, again: #ambitiousbeginner 

One of the circles lies with one half on the front of the pant leg, and the other on the back.  I wanted to make it seem like the pattern was continuing around the back of the leg.  I really like how it looks!

[thesimquarter]

Sims 2 Miniopolis Update

Ok. The second thing thing I'm sharing today is a Sims 2 Miniopolis update. I roughed out the last four builds I was planning to do for this... that being four builds that I really did not want to do. :)

under the cut! because i do not like making long posts

First up, the Rep group clubhouse... aka a non-euclidean nightmare of a building. I mean, ALL buildings don't have an inside that 100% matches their outside but this one... man

The windows look weird because each clubhouse calls for a different window type. I'm not certain about if I will normalize it or not. Probably should. None of the clubhouses are decorated either because... they're not decorated in-canon. That's YOUR job. I'm not sure if I'll decorate them later.

The Streeties clubhouse looked like a construction site to me, so I just made it look Purposefully Incomplete. For the vibes. The indescribable quality of Street.

Yes. Gorilla Statue as a placeholder. What else was I supposed to put there? The lumberjack?

Second up! Splicer Island.

I was originally going to made them into a bunch of smaller lots, but then I decided that's stupid and just made them one. The lab probably needs a bit more decoration, and the entertainment center has been interpreted, that's for sure. I replaced the pet show with a cooking contest because the booths look kinda similar to the tablecloth. Also, I needed to do SOMETHING with the cabana.

Maybe I should add a pet store here? Or in the Flea Market in The Sims Quarter... decisions decisions.

Third of all, the Brownstones/Slice O' Life!

I think this one turned out pretty well! There are three brownstones instead of just two because uhhhh... Sims 2 limitations. Slice O' Life had to fit under a building; it did NOT fit under just two. So, I made I three!

I'm thinking on adding a fourth brownstone... or doing something with the space up-top and turning it into a fourth apartment, but I haven't decided on anything yet.

And finally, the Carnival! Which I didn't want to do because there are pretty much no tools to make decent-looking carnivals in The Sims 2. …I will let the screenshots speak for themselves.

[irlactualhuman]

favorite word?

I don't really have a favorite. It's like music. I like music that goes hard, not really bound to genre or specific song. There are too many of either.

Obstreperous comes to mind. Fun to say. Very much enjoy a good O word. Like obfuscate or obsequious.

Anagram is a very pleasing word in construction and mouthfeel.

Torpid (being afflicted with a torpor) is good.

I like when simile is bound within and exemplified by a single word. Like Dionysian. Euclidean. Words that convey a feeling, image, and vibe all at once. If you're at all familiar with the concept, you can grip the feeling it's selling pretty damn easy.

Kinetic is just fun to say.

Logos. Logarithmic. L and G make very good friends in my brain.

I hope my rambling about words is enlightening. They are one of my favorite things.

[yamada-ryo]

Maybe if I skip the conditioner my hair won't turn into a topologically impossible non-euclidean construct this time

[ckret2]

Bill hates it when people mention Euclydia. Everyone thinks it's because he doesn't want to hear his home's real name; it's actually the opposite.

Here, have some fic. The naming of Euclydia (among other things), the birth of the Nightmare Realm, and the Axolotl planting the seeds of a trillion-year-long plan to keep Bill from the death penalty.

This is the 🎉FINAL PART🎉 of a 9-part plot about the Axolotl in the aftermath of the Euclidean Massacre. If you wanna read the others (or look at the art), here's one, two, three, four, five, six, seven, and eight.

####

With the immediate crisis averted and the triangle, for the moment, not attempting to invade and/or demolish the multiverse, most of the god militia pulled back. A group remained stationed near the unstable border between dimensions to watch the triangle; but the less powerful gods could trickle back in to get back to their own work, first and foremost the construction workers doing emergency repairs to reformat and stabilize the neighboring dimensions.

The Axolotl—who, he suspected, would have been arrested himself for interfering if they weren't still focused on the triangle—wove through the crowd until he found the Time Giant; and then swam angrily up to her and demanded, "You used me as a distraction?"

She turned a stone-hard look on him. "That was the agreement."

"No! The agreement was that I'd try to talk him down! We'd only resort to distracting him if I couldn't get through to him!"

"Ya didn't get through to him." The Time Giant nodded at the Axolotl's burned side. "Look at you. Your leg's off."

He looked down at his missing foreleg. He'd been so distracted by the near end of the multiverse, he'd barely noticed the pain. "It's just a flesh wound," he insisted. "I'm an axolotl, it'll grow back!"

She shook her head.

"I would have gotten through to him! You saw me talk him down after an entire army threatened him!" the Axolotl said. "What if I had succeeded, and when we left my tank he found out you already wrote him off?! You never gave me a chance—"

"We did give you a chance," she said testily, "and I saw that you weren't gonna succeed." She hooked a thumb over her belt and tapped a finger on her time tape; the stylized symbol of the Time Giants glowed on the side, an unsubtle reminder that she knew what was coming far better than he did. "So I did my damn job."

So she'd sent him in already knowing that he would fail. The Axolotl was speechless for a second. "But—you couldn't know—I got so close, if I'd had just one more try to talk to him..."

"If I'd let you, I'm sure you woulda kept trying until the end of time," she said. "You seem like a good guy, Ax—but you can't save everyone." She pushed past him to get to work. "There's first aid near where Dimension 2 Gamma was. Get those burns looked at."

"They're fine."

She was wrong. He could save everyone. Because he wouldn't stop until he did.

####

"You're replacing it?" the triangle asked petulantly.

"I'm not talking to you," VENDOR said, turned away from the triangle. "You had your chance at diplomacy and you blew it." The crablike cop was holding up a clipboard with some paperwork for VENDOR to review, and didn't look pleased to have been temporarily reduced to a secretary.

"I'm just asking a question!"

"We're not speaking."

At the top of his lungs—which was, it turned out, very loud and very shrill—the triangle said in the direction of the reporters, "Oh wow, that's a crazy thing to say about Lady Morgenstern! And talk about obscene! She'd be furious if she could hear that—!"

"Shhhhh!" VENDOR rounded angrily on the triangle. "You don't even know who she is!"

"I know her name and I'm not afraid to use it," the triangle said. "You're really replacing my dimension?"

"If I can be left alone long enough to finish signing the authorization paperwork," VENDOR muttered. "The construction crew's already out here and waiting, so if you don't mind..."

"It just seems pretty tacky, replacing a universe just like that." The triangle spoke like dimension he was talking about was just a pawn to be used in a trivial argument about etiquette, rather than everyone and everything he'd ever known. "No memorial or anything? Yeesh."

"So hold a memorial for it," VENDOR said. "We don't have any choice, we have to repair all the fallen walls to keep reality stable. If you'd let us into your hovel to sweep up what's left of your old dimension, it could have at least been incorporated into the new one."

The triangle half reached for his hat, stopped himself, and curled his hand into a fist and thrust it down at his side. "Over my dead body," he said. "Which I'm pretty sure got incinerated! So that means never!"

"You're pretty sure?" VENDOR asked archly.

"It... I had more important stuff to take care of, okay? I'm a busy guy!"

"I'm sure," VENDOR said. "Well, it's too late for any cleanup operations anyway. Enjoy rotting away in your landfill."

"Wow, that's how you talk to a refugee from the biggest disaster ever?" The triangle laughed. "Hey, bet the muckrakers over there would love to hear how sympathetic you are to the—what'd you say I am—the 'last surviving soul from my dimension'—?"

"Let's find somewhere quieter to work," VENDOR said to the cop.

He looked relieved "You got it."

As VENDOR and THEIR impromptu secretary moved away from Dimension Zero, the triangle shouted after THEM, "Hey! How do I vote for Municipalitron!"

Volcanoes on several of VENDOR's planets erupted. THEY whipped around to face the triangle. "You don't! You aren't in my district!"

"Well, whose district am I in? This Morgenstern creep you keep bringing up?" the triangle asked. "How's voting work, do you toss a ballot across the border and I toss it back—?"

"You're not in anyone's district! If you were, you'd have been arrested already!"

The triangle stared in dumb shock. "Wait, so I don't get to vote for which of you idiots I have to deal with?" He hollered at VENDOR's retreating back, "That's fascism!"

Fuming, VENDOR passed the Axolotl muttering under THEIR breath about showing the triangle fascism; then stopped, abruptly turned to face him, and snapped, "You."

"You," the Axolotl agreed.

"You're an optimistic fool."

Yes, well, he knew that already. He'd been voted Most Adorably Idealistic in his law school yearbook for a reason. "I don't think I like you, either."

"No one does." THEIR camera whirred irritably as they looked the Axolotl up and down. "What are you doing here, anyway? I assumed you'd been sent to figure out who's liable for this whole mess—but no, you only handle afterlife cases, don't you? Who sent you?"

The Axolotl was silent.

Furiously, VENDOR said, "Are you serious?! We could have avoided half this mess if it weren't for you!"

"If it weren't for me, he'd have knocked down the multiverse before anyone realized he's setting the fires," the Axolotl snapped. "And if you had figured that much out, you'd have gotten your cops killed before anyone realized he's a god."

"The professionals here to handle the situation could have figured it out faster if you weren't derailing their investigations," VENDOR snarled. "And arguing about jurisdiction! We could have arrested that that little troublemaker the moment we figured out just what he's done—"

"Right after you arrested that kid with the spray can who didn't have anything to do with this?"

THEY growled in frustration. "Forget it! I hope you're happy with your genocidal pal over there—you seem about as concerned with public safety as he is." THEY stormed off, the cop with THEIR paperwork chasing after THEM.

The Axolotl watched VENDOR go; then turned to look ruefully toward Dimension Zero.

When the triangle caught his gaze, he formed a heart with his fingers over his top point and called out, gleefully singsong, "Genocide paaals!"

It wasn't exactly the reaction he'd hoped for.

####

The Axolotl was attempting to distract himself from scratching his itchy leg while it regrew by eavesdropping on the triangle. It seemed like the triangle was entertaining himself by darting around the border of Dimension Zero to start arguments with anybody he happened to recognize (except the Axolotl, whom he seemed to be trying to ignore outside of throwing a few odd quips at him.) At the moment, the triangle and the Time Giant were hollering at each other about her decision to reinforce the second dimensions by making them splinter into multiple timelines.

"So you're really willing to sacrifice zillions of lives by letting me incinerate all their parallel timelines?" The triangle laughed in disbelief. "And everyone here thinks I'm the killer! That's not a good look for you, buddy!"

She glanced up from a table full of paperwork to give him a totally neutral look. "You're the one who's willing to incinerate them. You could not do that."

"When I do it, it's justified."

The Axolotl was distracted from the argument as the storm cloud with the apoc agents gloomily blew past him. It was talking into a walkie-talkie as it went: "Yeah, I know he's a nut. But he's a nut that can't throw fireballs outside the border of his dimension, and I've got to finish this report before we can get outta here." He sighed at whatever the walkie-talkie said in response, and said, "Yeah. We'll rendezvous after I have his testimony." It let its tornado suck the walkie-talkie back in and drifted to the Time Giant. "Mind if I steal your conversation partner for a minute? ATTF business."

She grabbed a binder to try to shield her papers from the worst of the storm's rain. "Please. Take him."

"Thanks." It floated closer to Dimension Zero and raised its voice to bark, "Hey! Magister Mentium!"

The triangle looked over mistrustfully. "What?" As he'd talked to the Time Giant, he'd been playing with the fabric of reality, creating a circle out of raw... stuff. The Axolotl couldn't tell what the stuff was, but it looked like it was some sort of animal tissue, except far too uncannily homogeneous to be natural, disturbing in its uniformity. Like a slice of baloney. When he saw who'd called out to him, he rolled his eye and turned his attention to extruding the circle into a baloney cylinder. "Heeey, Officer Fun Police! Here to rain on my parade again?"

"Rain jokes aren't as funny as you think they are," it said. "No, this is Apocalyptic Threat Task Force business."

The triangle's eye narrowed. "What business? Are you gonna complain about my renovations again?"

"No. If you're not about to knock reality down, I don't care what you do anymore," the cloud said. "It's not my business to punish anybody for previous apocalypses, I just want to prevent future ones. Answer a few questions for our incident report and I'll be out of your life." There was an implicit and you'll be out of mine in its tone.

"All right," the triangle said dubiously. "Fffine. Then we're on the same side. I'm not fond of apocalypses either."

It paused like it wanted to argue with that claim, but said, "Good enough for me." It pulled out the soggy notepad it had been using all day, flipped through it, couldn't find a free page, and with a sigh pulled out a tape recorder instead. "You're from Dimension 2 Delta, right?"

"If you say so," the triangle said, lifting his hands in a shrug. "You guys are the ones who named my dimension."

"Uh-huh." Under its breath, the cloud muttered, "Not exactly a name, but... If you're from 2Δ, that makes you the only direct witness to how your universe was destroyed."

The triangle paused. "Mm."

"Can you explain what happened, exactly?" When the triangle didn't respond, the cloud added, "I'm not gonna arrest you for it. If we want to have a chance of stopping something like this from happening in the future, we need to know what happened here."

"Uhhh, yyyeah. Suuure," the triangle said.  It wasn't clear exactly how Dimension Zero rearranged, but the view of the eternal dance party simply vanished. There was no sign of the millions of shapes. The music had fallen near silent, just a constant distant low thumping noise, like your heartbeat in your ears; quiet enough that it couldn't drown out the whispery hiss leaking out of Dimension Zero. "It's not like I have anything to hide." Whatever he was about to say, it seemed like he wanted to hide it from his party prisoners, at least.

A bolt of lightning shot through the storm's recorder, turning it on. "You said you were an active participant in the end of the world, right?"

"Hey, what's that supposed to mean?" He eyed the recorder suspiciously. "What is this, some trick to try to get a confession out of me?"

"Again, I'm not a cop. And you already confessed in front of a thousand reporters," the storm said. "If you were involved, you've got a different perspective than some guy ten superclusters away who only witnessed it, that's the only reason it matters."

"Oh," the triangle said. "Then—yeah, I was there for the whole thing. Start to finish."

"Great," the storm said gruffly. "Then could you explain in your own words what happened when the universe ended and, to the best of your knowledge, what caused it."

"Oh. Yeah. Right. The cause," the triangle said. "It... it was a—monster."

"I thought you said you—"

"It was a monster," the triangle said, more confidently now.

The cloud hesitated. "All right," it said. "Tell me what happened."

The triangle took a deep breath. "Okay. So. It uh—started with the third dimension."

"The monster came from the third dimension?"

"No, we were going to the third dimension. But we needed—"

The hissing background static exploded into a roar.

The void filled with the staticky screams of countless dead voices, pleading for mercy, pleading for it to stop. Death rattles, howls of agony, wails of terror. Most of the crowd of gods outside Dimension Zero fell silent, turning to stare at the disembodied hysterical shrieks.

One voice, strained with pain, rose above the cacophony, crackling, "Emergency services! We need medical assistance! Ambulances, or—please—I don't know what happened—it's like everyone's internal organs spontaneously ruptured, there's—there's hundreds of people here! Some of them are missing parts of their body, they just—disappeared! I'm hurt too, I don't know what it is—I can feel it inside me—"

A second voice replied, "We can't send assistance. Everyone's bleeding, the whole city's dying! We can't help you!"

Whatever the triangle said was lost beneath the roar. He didn't even seem to notice it. His eye was filled with static. The word "blood" was just barely audible. The word "mandibles."

Another voice, trying to sound professional, trying to sound authoritative, but trembling with fear, "This is an emergency announcement! This announcement will not repeat! The fire can transmit over radio waves and sound waves! Turn off all radios and TVs! Turn off all radios and TVs and destroy any wireless phones and pagers! Do NOT listen to the screams! Again, the fire is transmitting over radio waves, this message will not repeat, destroy your radio and warn your neighbors!"

The Axolotl saw images flash in the triangle's eye, too fast for him to mentally process one before another ten had gone by: a plane like infinitely thin glass with tiny delicate shapes painted on its surface shattering in a rolling wave; a bleeding body reduced to shards and then the shards reduced to chips and then chips reduced to dust; fire spitting and crackling into every crack split in existence; a light shaped like a triangle. (Was that the light that had blinded the Oracle's seer?)

Another voice gasping, "It's doing something to the gravity, I-I don't understand—we don't even have the equipment to read... it's like gravity's turned in a direction that doesn't exist! Does anyone know how to stop it?! Our universe is tearing ap—" and the words were cut off with a scream; and the scream was cut off with a sudden silence that was swallowed whole by the other voices.

The triangle had peeled open, shining golden panels stretching out like petals, his mandibles unhinged and curling around his eye in a ring of teeth, like a blooming carnivorous flower, sun-soaked and mesmerizing. God, he was so bright. He shot light in every direction like an explosion that never ended. Like a star trapped in the moment of supernova.

Another voice, shaking with rage, "Did you hear that, you monster?! I told you we weren't ready yet! Why didn't you listen?! I can see the destruction from here—the sky's on fire, everything is burning. How could this happen?! YOU killed them all—" and the rage cracked, revealing the fear and grief just barely hidden underneath, "Remember us. If you're the only one left, you have to remember us. Please—"

The static snapped off; the triangle's body snapped back into place; his eye snapped back into focus; "—and then they appointed me their god," he said cheerfully, "and here we are!"

And with only a couple more dying cries of pain and pleas for help, the voices fell back to their constant background whisper.

The storm cloud had started sleeting.

The Axolotl had stopped breathing. Just the sound of the carnage was enough to make him sick.

But the triangle sounded perfectly at ease—more than he had before he'd answered the cloud's question. "So is that all you needed?" He'd resumed playing with the cylinder of meat he'd been constructing—extruding it further, and then, dissatisfied with the results, collapsing it back into a circle.

His hands were trembling as he messed with the cylinder. There was a tightness around his eye.

"What..." The storm cloud let out a low rumble of thunder, ahem, "what... did you say about blood? I didn't catch it."

The triangle blinked blankly at the storm. "I didn't say anything about blood."

It paused.  "All right, then—what about the other voices? Who were they?"

"What voices?"

The storm stared at the triangle, baffled sunbeam fixed on him; then swung the sunbeam over to the Axolotl. "You heard—?"

So his eavesdropping had been noticed. He nodded. Oh, he heard, all right.

The triangle glanced between them. "I think you guys are hearing voices," he said. "The only one talking here is me."

He said it like he meant it. The Axolotl was sure he did. Had he not heard the voices?

"Never mind, forget it," the cloud said uneasily. "You said someone... Who appointed you their god?"

"Uhhh..." the triangle tilted to the side as he tried to think. "Pretty much all my people? Yeah. It was everyone!"

"Your people? From your universe?"

"Yup!"

"They didn't appoint you their god," the cloud said. "They're all dead."

The triangle scoffed. "I don't know what you're talking about. They're all in here with me!"

"You mean the mortals from the other universes?"

"I don't know what you're talking about," the triangle repeated, a little slower, warningly. "They're all from my universe."

For a moment, the cloud just stared at him, at a loss. It glanced again toward the Axolotl. The Axolotl had nothing to offer it.

"Is that everything?" The triangle tried to keep his voice peppy, but there was an edge of exhaustion that hadn't been there earlier. (Yeah, him and everyone else here.)

"I guess that wraps up that part of the questionnaire," the cloud muttered uneasily, trying to recover its professional tone. "Just a couple more questions. I need your name. For the report."

Dimension Zero's hissing background static rose again: "The murderer... The name of the murderer... is—"

"NOBODY ASKED YOU!" The triangle turned and chucked the cylinder he'd been working on into the Dream Realm. He grumbled under his breath, created another circle, and started stretching it out again.

The triangle could hear the voices. Then why hadn't he been able to hear them earlier? Unless he had been able to hear them—and he just... couldn't remember that he'd heard them?

Even if the Axolotl hadn't known about the incomparable trauma the triangle had survived/caused, it would be pretty obvious by now that something was going terribly wrong inside his head. Contradictory stories about his own reality, memories he refused to remember, facts he simply set aside as not relevant. Was he refusing to face them, or was he unable?

From their conversation in the Axolotl's tank, he thought the triangle understood more than he was willing to admit. But the Axolotl might be the only one who knew that.

And that was beginning to give the Axolotl an idea.

"Just—put me down as the Magister Mentium, okay?" the triangle told the cloud. "Everyone'll know who you're talking about."

"If you say so," said the cloud. "What was your universe's name?"

"Its name?" The triangle glanced up from his new cylinder and gave the cloud a perplexed look. "You asked already. You said it's Dimension 2 Delta."

"That's its serial number. Every dimension's assigned one at its Big Bang. But it's standard to let a dimension's own residents choose its name. It makes it more personal." The cloud sounded as though it had memorized this explanation. The Axolotl wondered how many times it had had to take statements from a destroyed dimension's grieving survivors. He hoped it usually got to give this spiel to witnesses of a narrowly averted apocalypse. "Typically the first explorers to leave their dimension get to name it; but the only person ever known to leave 2Δ is... you."

"Oh," he said. "Right."

"So, what did your people name your universe?"

He stared at the storm like it was stupid. "We called it... the universe?"

"Everyone calls their universe The Universe," the cloud said. "Followed by The World, The Dimension, Reality, and Home. They're all taken, come up with something else."

"Seriously? You're making me name my whole universe and now you're telling me how to name it?"

"They're not my rules," the cloud said. "If you don't have a native name, we usually name a dimension after the first known explorer to leave it. Was that you?"

The triangle was quiet for an uncomfortably long moment. His gaze twitched away; and for a moment the Axolotl thought he saw another image flash in his eye: a triangle floating in space, eerily serene, dead. His voice was small when he said, "No."

Surprised lightning quietly flashed in the storm's cloud. "Oh. Do you know the name of the first?"

"Of course I do. He's my..." He stopped himself. He said, too evenly, "His name is Euclid."

Obviously, the triangle wasn't speaking a language that can be spoken with human mouths or written with human symbols. "Euclid" is a stand-in word for an unpronounceable name; trying to say the name without the right anatomy—without even the right laws of physics and sound waves—would only mangle it.

But the rest of the multiverse didn't have the right physics or anatomy either. "Euclid," the cloud repeated, mangling it. The triangle winced. "Fine. How's Euclydia sound?"

"It sounds stupid," the triangle said.

"Well, it's your dimension. Do you have a better suggestion?"

"I..." The triangle floundered helplessly. "That... Okay hold on, I've had a very long..." He floundered again as he tried to figure exactly what kind of time span he'd been having a long one of.

"If you want me to come back later..." said the cloud, who very obviously did not want to have to come back later.

"I don't knowww, gimme a second," the triangle whined. "I've never thought about a universe having a name! It's—it's fine. Euclydia's fine."

"If you're sure—?"

"Of course I'm sure," the triangle snapped. "Euclydia. Yeah. Great. Fine."

"All right." The cloud zapped its tape recorder, turning it off. "Thanks for your time."

As it started to hover off, the triangle said, "Hold on! I answered your questions, you owe me some."

The eye of the storm reluctantly swung back toward the triangle. "What?"

He held up the shape he'd been extruding. "What do you call this... 3D circle thing?"

The sunbeam swept over it. "A cylinder?"

The triangle pointed toward VENDOR, who was out at the edge of the crowd answering the questions of some reporters who'd caught THEM attempting to slink away from the scene. "And what are the 3D circle things Coin Slot over there is hauling around?"

It glanced at VENDOR's stock of planets. "Spheres."

The triangle shook his cylinder. "Well, what am I doing wrong, then!"

"I don't know, math's not my thing," the cloud said. "Try rotating it."

The triangle waited until the cloud had moved on; then created another circle, extruded it again, but curled the extrusion around into a circle. He ended up with a shape like a donut. He said, quietly, "Oo-oo-ooh." He sounded impressed.

The Axolotl swam up alongside the storm cloud as it left. "So. Find out what you wanted to know?"

The cloud laughed ruefully.

That was what he thought. "Are the interviews you've been taking classified?"

"No, our reports are open to the public. Anyone can request copies. The database is a nightmare to navigate, though."

"Let me know who to contact for the records on this incident. Especially the witness testimonies."

"I take it you're also planning to go through that noise we just heard with a fine-tooth comb?"

"That's hardly the start of it."

If the Axolotl had been convinced of anything during all his conversations with the triangle today, it was that the triangle could barely begin to grasp just what it was he'd done to his dimension and all the dimensions around it—and he did a very poor job of communicating what he did grasp.

And if the Axolotl could prove that—if he could build a convincing argument that the triangle hadn't understood what he'd done, psychologically couldn't understand, that even now he only had the fuzziest comprehension of what he was involved in...

Someday, that triangle's sins would catch up to him. Someday, he would be in the hands of the gods of death and justice, and they would have to decide what fate his actions had earned. And when that day came, it would be the Axolotl's job to ensure that the triangle didn't end up damned or erased from existence.

As it was now, that triangle didn't stand a chance in the multiverse of being found innocent. But there was more than one way to avoid a "guilty" verdict.

By the time the triangle stood before a judge, the Axolotl would make sure that the right laws were in place for him to do what he wanted to do.

####

Where there had been swarms of firefighters earlier, now the scene swarmed with construction workers, working on the emergency genesis of over half a dozen replacement universes—carefully, so that the big bangs didn't do any further damage to an already unstable situation; but quickly. Already every destroyed one-dimensional universe had been replaced. Several half-burned dimensions had been supplanted with oddly-shaped undersized universes that met at the older universes' burned edges; jagged 1D dimensions sealed the gaps between these dimensions like a line of solder between two panes of stained glass.

By now, the flat planes and edges surrounded the zeroth dimension like the sleek shifting surfaces of an infinity-sided die; all except for one last missing wall in the middle of the damage.

Dimension 2 Delta. "Euclydia."

The construction workers were already setting up the scaffolding and equipment to set off another big bang.

As the Axolotl looked at the copious warning signs around the construction site—"DANGER! COSMIC EXPLOSIVES" "GENESIS IN PROGRESS"—the specialized equipment, the veritable army of workers, the mountain of papers the Time Giant had been reviewing earlier to ensure that everything was up to code and nothing would go wrong... he couldn't help but think of the triangle holding the seed of a big bang in his bare glowing hand, threatening to set it off right there. The Axolotl had known it was foolish, but seeing all the workers' preparations put just how reckless it was into perspective. Like a toddler holding a stick of TNT over a campfire.

He spotted the Time Giant among the workers, flickering back and forth across the scene as she tried to literally be multiple places at the same time. When she settled down for a moment over a worktable to double check a pile of blueprints and forms and calculations and even more paperwork, she caught sight of the Axolotl passing by, and tipped her chin up at him in greeting.

He paused, then nodded back to her. No hard feelings. He was just following his principles; and she was just doing her job. They'd each found their own way to help hold up the multiverse.

"Hey," she called out, and gestured for him to come over. As he did, she said, "Your leg's healing nicely."

He glanced down at it. His new toes were stubby, but at least they were back. "I don't like being uneven." He'd take a few more days on his tail. "I'll probably pay for it tomorrow, though." When he finally got home, he'd have to see if he could cancel his morning appointments.

"Reckon we'll all be feeling this tomorrow." She tilted her head toward Dimension Zero. "I've got a message for the god of DIY over there. I think you're the only one he likes—you mind carrying it over?"

####

It wasn't hard to find the triangle; he was leaning against the membrane around the zeroth dimension, moodily staring out at the third. He seemed to be gazing past all the gods, unfazed by their hubbub. The Axolotl tried to see what he was looking at, and didn't spot anything of note. As far as he could tell, the triangle might as well just be stargazing.

Along with the police tape and the ATTF barrier and the long-forgotten cordons to hold off the reporters, there was now an additional grid of orange cones set up blocking anyone from getting too close to the destroyed wall and the construction site. The Axolotl glanced around to make sure no one was paying attention before he slipped past the cones and swam up to the triangle.

When he approached, the triangle was muttering under his breath: "Stupid, now it sounds like an STD. I should've named it something cooler. Like... Triangletopia. Or the Party Plane. Or Margaritaville—I bet no one's ever used that one before..."

"Magister," the Axolotl said.

The triangle's eye snapped to him. "Hey, look at that! The pompous psycho is back! If you're even thinking about sticking me back in your 'office'—"

The Axolotl held up his forelegs appeasingly. "I'm not." He wasn't even crossing the threshold into the triangle's turf. "This is the last time I'll speak to you today."

"Finally, some good news," the triangle grumbled. "What do you w—ha! Ah-haha! I caught myself, that one didn't count."

The Axolotl decided not to count it. "The Time Giant wanted you to know they're about to set off the big bang where Dimension 2 Delta used to be. You probably don't want to be too close to the wall when it goes up."

The triangle's expression darkened; but he just said, "All right. Fine. Have fun. Not my problem! Just keep the construction noises down."

That was all he'd been sent to tell the triangle; but he added, "If you ever want to leave your dream realm, this is your last chance."

The triangle groaned. "This again? Listen, frills, I already told you I'm not interested! And you don't have the right to drag me out, this is my sovereign god territory—"

"I'm not threatening to," the Axolotl said gently. "I just—wanted to make sure you know. If you change your mind later, you physically won't be able to leave."

That gave the triangle pause. "I... don't see why not."

"For something to pass from one dimension to another, it needs a large enough hole to pass through," the Axolotl said. "For a person carrying the mass and energy of an entire universe to cross from one dimension to another... they need a hole the size of a universe. The missing wall where 2Δ was is the size your universe used to be. And now... it's the only exit big enough for you to pass through. Do you understand?"

The triangle stared at him silently. There was that hard, heavy look in his eye. It was awful to see. He did understand.

"If you don't come now..."

"We came up with a way to fit my entire universe into this one," the triangle said. "If I ever want to leave, we'll invent a way to get it back out."

"Your universe didn't fit in without incinerating it."

The triangle tapped the side of his hat with a finger; somewhere inside it was the speck that used to be his universe—the seed of a big bang. "It's travel-sized now. The next time will be easier."

For the first time since seeing the awful ruin of Dimension 2 Delta, the Axolotl forced himself to turn his fearful gaze chronologically forward. He squinted toward the hazy, far-flung future; and then he gave the triangle, in the present, a sorrowful look. "No, it won't," he said. "But I'll do what I can for you."

The triangle stared sullenly at him, unmoved by the offer. "I don't see what you're getting out of helping me. Everyone else is dying to send me to ghost jail or however things work around here."

"Isn't it enough to help you just because you exist and that makes you worth it?"

"If you ever, ever say something like that again, I'll kill you. I will find a way."

He wasn't particularly surprised. But that was truly what the Axolotl believed—and believed strongly enough to guide everything else he did. 

The things this triangle had done were too ghastly for even an ancient, experienced god to fully wrap his head around. Without exaggeration, he might have done the worst thing anyone anywhere in the multiverse had ever done.

But.

But if the Axolotl could prove that he, the worst person ever, was worth giving a second chance—that he could change, that he could show remorse for what he'd done, that he could be a force for good in the multiverse... then he would have proven that everyone, no matter what, was worth it.

The Axolotl had been voted Most Adorably Idealistic, but he'd never been called soft. His ideals were harder than diamond and sharper than obsidian. He hadn't decided to protect the triangle in spite of the impact that might have on the multiverse; he was protecting him because of the impact it could have. 

The Axolotl was a god of justice, of monsters, of second chances, and through his actions he could shape what justice meant throughout the multiverse as if he were sculpting clay; and he thought a small, sharp little equilateral triangle would make a perfect sculpting tool.

"In truth, I just don't believe in punishment. Not even for you." The Axolotl lay a forefoot on Dimension Zero's bubble. "But I don't see why you trust me." Because it was clear the triangle did. He'd trusted the Axolotl to judge the character of the other gods. He'd kept looking toward him like he was trying to gauge his own situation based on the Axolotl's reaction to it. He'd admitted the truth about the remains of his universe and his plans for it. It seemed like the Axolotl was the only one the triangle trusted in all this mess.

The triangle thought that over; then said, "You seem like a grade-A sucker."

He laughed. "I'll try to live up to your opinion of me." He had a guess what kind of people this triangle thought were suckers. The charitable; the caring. The people who didn't think that seeing the worth in everyone was a kind of illness.

"You should know, I intend to legally register my tank as a purgatory. I'll probably submit my application before the end of the week. If you claim it as your afterlife, you'll be transferred to my tank for holding while awaiting trial to decide your final afterlife."

"Ugh, now it all makes sense: you're starting a cult! I don't wanna join your cult, frills—I've got my own."

"But you do want to go straight to your lawyer's office if you're about to go on trial for your sins," the Axolotl said pointedly. "I don't intend to house anyone in my tank permanently. It will just be a transfer place for clients preparing for trial or figuring out where they want to go next—another afterlife, reincarnation... You're already technically dead; you can request at any time to come to my tank, and you'll be there."

"Sounds great for your other clients! But I'm not planning to go on trial and I don't want to be in an afterlife," the triangle said testily. "I'm pretty sure we've been over this!"

"I know you don't. I wish you didn't have to face it. But when you have no choice," the Axolotl said. "When you need it. When your time comes to burn like your people—" (the triangle flinched) "—call me. I'll offer you a second chance at any time."

"Low blow," the triangle muttered. "Don't put yourself out on my account. I'll be fine by myself."

"I'm sure." The Axolotl suspected he'd be putting himself out on the triangle's account for a long time. "What's your name? Your real name."

The background hiss of cosmic noise roared louder. The echoes of billions of erased ghosts said, "THE NAME OF THE MURDERER IS—"

With a flinch, the triangle cranked the distant dance music louder so it spilled cacophonously out of Dimension Zero again. It was too late, though. The Axolotl had heard the triangle's real name.

He pretended he hadn't. He waited.

The triangle didn't answer for a long moment. "You probably wouldn't be able to pronounce it."

"Maybe not." He'd seen how the triangle had winced hearing the cloud try to pronounce the name of some other shape. "I still want to know who you are."

He wrestled with his words; then finally gave up and asked his question. "What... is this place? We're not in the third dimension. When I—freed my dimension, I expected to go up; but we went... down. I didn't know there was a down." He confessed his ignorance in a near whisper, almost drowned out by his own music.

"You're in Dimension Zero." But that wasn't right. Dimension Zero was—should be—a point, and it's impossible to be "in" a point. A point simply is. "You are Dimension Zero."

The triangle said, "Then call me King Zero."

The Axolotl considered that. "Yes," he said. "I think that is your name."

Someone shouted, "Clear the way!" One worker at the construction site was looking directly at the Axolotl. "That means you! Unless you wanna be boiled frog legs!"

"I'm not a frog," the Axolotl muttered; but, he turned one last time to newly-crowned King Zero, said, "Call me," then hastily swam to the safe side of the orange cone barricade.

"Five, four, three..."

The Axolotl watched the triangle—and the triangle watched him—until the detonation. The big bang went off in a flash of light bright enough it would have incinerated anyone in the vicinity had it not been contained to a flat plane.

When the Axolotl looked away from the light, the afterimage of a triangle was burned into the center of his vision.

Dimension Zero was sealed off from the rest of reality—locking its king in for the next trillion years.

####

When the triangle said his name was "King Zero," of course, he wasn't speaking English. English wouldn't exist for a long time. The name King Zero is simply a convenient translation.

The English word "zero" comes from the French zéro. Zéro comes from Italian zefiro. Zefiro comes from Medieval Latin zephirum. And zephirum comes from the Arabic صِفْر—ṣifr.

####

Centuries ago, in the dream of a naive, trusting human, the human asked in Arabic, "What should I call you?" And King Zero responded, "Call me Ṣifr."

And years later, a dreaming human asked in Medieval Latin, "What should I call you, o muse of mathematics?" And of the two Latin words descended from his current Arabic nickname, Ṣifr responded with the one he thought was closer: "Call me Cifra."

A dreaming human asked in Old French, "What's your name?" And he replied, "My name's Cyffre."

Speaking Middle English, he told a dreaming human, "My name's Siphre."

And in Modern English, he told Edward Bishop Bishop, "The name's Cipher. But you can call me Bill."

In a year's time, and two years before his death from sleep deprivation, Edward would write Flatworld, a book about a 2D shape and his Muse journeying up to the highest dimensions; and also all the way down, below the spaces and planes and lines, to the self-absorbed King Zero, buried in the point-sized zeroth dimension, who thought a whole universe was contained inside him.

####

(It's FINISHED. 🎉🎉🎉

Hi y'all, if you just joined us for this Axolotl plot arc, usually this is a post-canon human Bill fic. I took a break from the main plot for one week to post a one-chapter flashback and then it was nine chapters. This bitch is 50k words. It's a novel unto itself.

Anyway if you only showed up for this story about the Ax, it only exists in service of a much longer story; so if you enjoyed this check out the rest of the fic. This is technically chapter 69 (lol). (If human Bill isn't usually your thing, I've been told that this is The Human Bill Fic For People Who Don't Like Human Bills because Bill is clearly very much a triangle unhappily trapped in a human body, rather than just chill with being human—so you might wanna give it a shot.)

And for the regulars who are already reading the whole fic: OH MY GOD IT'S FINALLY FINISHED, WE'RE FREE, WE CAN RETURN TO THE PRESENT. Listen I love the Ax and his bizarre but unbending morality, but guys. Guys. I miss Mabel so much.

Pre-warning that I may end up needing to skip a chapter or two before the end of the year, because work's piling a LOTTA extra work on me this month and I might just flat out not have time to edit & do art. I'm up at 3 a.m. editing & queueing this post and I was up til 3 a.m. another night doing the art because I HAVE NOT HAD TIME this week to do it any earlier. I did this because I love y'all.

No that's a lie, I did this because I want to FINISH this DANG ARC. That's my birthday gift to me.

Anyway lemme know what y'all think!! 💕)